Magnitude

A vector is comprised of two components: magnitude and direction. The direction of a vector refers to the imaginary rotation that is necessary to move the object from some given reference point to its current position in space. The magnitude of a vector, v, is its absolute length, measured between the tail and head of the vector. Magnitude provides a reference for the size of vectors (or other mathematical objects) relative to other vectors, and is denoted ||v||. Another name for the magnitude of v is the Euclidean norm of v, in honor of Euclid, one of the first mathematicians to do serious work concerning the geometry of length, distance, and angles.

To find the magnitude of an n-vector,

use the following formula:

where v_n represents the components of the vector; given that the point A = (x₁, y₁) is the tail end of the vector v, and B = (x₂, y₂) is the terminal end, the components of vector v are:

For example, given the tail end of vector v is at point (7, 3) and its terminal end is (12, 10), the components of vector v are

and vector v can be written in component form as:

Properties of vectors and magnitude

There are certain properties that are useful to know when working with vectors and magnitude. One of these is the effect of scalar multiplication on the magnitude of a vector.

If v is a vector and c is a real-number scalar, then the magnitude of the scaled vector cv is given by:

This should make sense because if we stretch v by a factor of c, then the length of v should be stretched by a factor of |c| since ||cv|| measures the absolute-value length of vector cv, as depicted in the figure below.

Another concept worth mentioning is the zero vector. The zero vector is, as its name implies, a vector that has a magnitude of 0. It is also referred to as the null vector. Because the zero vector has a magnitude of 0 it follows that it also has no direction, since it has no length, and all of its components are equal to 0. Thus, the zero vector in 2-dimensional space can be written as:

The zero vector in 3-dimensional (and n-dimensional) space can be represented similarly:

Applications of vector magnitude

One of the most important applications of vector magnitude is normalizing a vector. Normalizing a vector involves keeping the direction of the vector while changing its magnitude to 1, which converts it into what is referred to as a unit vector. Since we saw above that a vector, v, multiplied by a scalar, c, simply has a magnitude of cv (and does not affect the direction of the vector), converting a vector to a unit vector allows us to work much more easily with vectors because the direction of the vector becomes its most important defining characteristic; we can just factor out the scalar and work with unit vectors. To convert a vector into a unit vector, divide each of its components by its magnitude such that,

where is the unit vector of vector v. Referencing the example in the section above, we found that the magnitude of vector v = [5, 7]^T is 8.602. Thus, to convert v to its unit vector, divide each of its components by 8.602: